Can AI research lead to new findings in general cognitive science?
This is about hard AI and soft AI: proponents of hard AI work on systems that simulate the way human cognition works, with the eventual (hypothetical) goal of replicating it. This presupposes that you know how cognition works, and presumably you will learn about it as you attempt to replicate it.

Soft AI, on the other hand, tries to emulate the outcomes only. For example, Weizenbaum's ELIZA is clearly on this side, as it uses simple pattern matching, and does not 'understand' anything about the conversations it is having.

Obviously, we don't even know fully what it means to 'understand' something, and building working systems is not really possible with a hard approach. Hence, soft AI is more common, as researchers are usually measured by their outcomes rather than their ideas. As far as I am aware, the hard AI approach has been all but abandoned long ago.

As current AI seems to be dominated by statistical approaches, I doubt that we can find out many useful things about cognition this way.

One interesting side-note: it seems to me that the capabilities of modern AI systems have developed away from human capabilities. A three-year-old can do some things that a sophisticated AI system cannot do, but in some areas (chess, translation, ...) the capabilities of AI systems surpass what humans are capable of. Maybe imitation is indeed not the right way to approach AI.